Theorem bnj1071 34953

Description: Technical lemma for bnj69 34986. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypothesis

Ref	Expression
bnj1071.7	⊢ 𝐷 = (ω ∖ {∅})

Assertion

Ref	Expression
bnj1071	⊢ (𝑛 ∈ 𝐷 → E Fr 𝑛)

Proof of Theorem bnj1071

Step	Hyp	Ref	Expression
1		bnj1071.7	. . 3 ⊢ 𝐷 = (ω ∖ {∅})
2	1	bnj923 34744	. 2 ⊢ (𝑛 ∈ 𝐷 → 𝑛 ∈ ω)
3		nnord 7911	. 2 ⊢ (𝑛 ∈ ω → Ord 𝑛)
4		ordfr 6410	. 2 ⊢ (Ord 𝑛 → E Fr 𝑛)
5	2, 3, 4	3syl 18	1 ⊢ (𝑛 ∈ 𝐷 → E Fr 𝑛)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2108   ∖ cdif 3973  ∅c0 4352  {csn 4648   E cep 5598   Fr wfr 5649  Ord word 6394  ωcom 7903

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rab 3444  df-v 3490  df-dif 3979  df-ss 3993  df-uni 4932  df-tr 5284  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-ord 6398  df-on 6399  df-om 7904

This theorem is referenced by:  bnj1030  34963  bnj1133  34965